From the eigenvalue problem Eq. (REF ), we know that
the completed set contains the branch of the negative frequency.
And from the topological nature of the PHE [1]},
the formula of the phonon Hall conductivity can be written
in the form comprises the contribution of all the branches
including both positive and negative frequency branches.
In order to simplify the notation, for all the branches, we define
\(a_{-k}=a_{k}^{\dagger }.\) 
